I bought a used HP Omnibook XE2 DC. It presently has a Celeron 450Mhz processor. I've seen an 800Mhz processor on ebay. Same bus speed (100Mhz), same socket type (MPGA2). If I put it in will it make this notebook faster, or just too hot?
I'm presently building a new desktop so a new laptop will have to wait.

I woudn't attempt any upgrade on a laptop other than memory and I think that you wouldn't have even asked this question if you saw how hard it is to open up a laptop for that kind of upgrade. Also, being an HP I know how difficult it is to open even a desktop. You would notice a speed boost, but then again the amount of added heat might just slow it down canceling each other out. How much is the cpu you are looking at? It might just be better if you maxed out your ram instead.

For all the pain of doing this 'not for the novice' exercise, plus the risk of ruining a working Laptop and a new Processor, go and buy yourself a new model laptop from an online auction.

You really need a full user / service manual and I have only seen these available from Compaq [not HP]. This would give you the part numbers for the devices, such as Motherboard and CPU.

Laptop CPU's do not come as discrete components. The are usually mounted onto a sub assembly, complete with cooling fan, heatsink etc.

You will also need to identify any jumper settings that require to be altered. On top of that, just the simple process of buying a 'processor module' will not be an easy exercise.